Criteria for the Absence and Existence of Arbitrage in Multi-and Infinite-Dimensional Diffusion ...

37 Pages·2016·0.47 MB·English

CRITERIA FOR THE ABSENCE AND EXISTENCE OF
ARBITRAGE IN MULTI- AND INFINITE-DIMENSIONAL
DIFFUSION MARKETS
DAVIDCRIENS
6
1
0
2 Abstract. Wederiveabstractaswellasdeterministicconditions fortheab-
senceandexistenceofarbitrageandfinancialbubblesinageneral(multi-and
v
infinite-dimensional) semimartingale-diffusion markets, and a Heath-Jarrow-
o
Morton-Musiela framework. We also provide deterministic conditions for the
N
martingale property of stochastic exponentials which are driven by solution
togeneralizedmartingaleproblems,respectivelystochasticpartialdifferential
7
equations. As an application, we construct a financial market in which the

numberofassetsdeterminestheabsenceofarbitragewhilethesourcesofrisk
]
F havethesamedimension.
M
. 1. Introduction
n
i Thequestionwhenafinancialmodelisfreeofarbitrageistypicallyaskforeachfi-
f
- nancialmodelindividually.Ourgoalistoprovideasystematicdiscussionforclasses
q
of models driven by multi- and infinite-dimensional semimartingale-diffusions, re-
[
spectively solutions to stochastic partial differential equations (SPDEs).

2
Letusexplainoursettingandaimsinmoredetail.Ononehand,weassumethat
v
1 the discounted price process S =(Si)i∈K is given by
2 dSi =Si e ,dX , Si =1,
6 t th i ti 0
1 where X is a semimartingale-diffusion with values in a real and separable Hilbert
0
. space with orthonormal basis (ei)i∈K. We define X in terms of a generalized mar-
9 tingale problem, which extends the martingale problem of Stroock and Varadhan
0 [86]toHilbertspacesandintroducesthe possibilityofexplosion.IfX isRd-valued,
6
wecanonicallyrecoverclassicalfinite-dimensionaldiffusionmarkets.Forthisframe-
1
: work we study the notions of no free lunch with vanishing risk (NFLVR), c.f. Del-
v
baen and Schachermayer [14, 18], no generalized arbitrage (NGA), c.f. Cherny [8]
i
X and Yan [93], and no unbounded profit with bounded risk (NUPBR), c.f. Karatzas
r and Kardaras [56, 53], and Kabanov [48]. This is done by considering the sets
a
ofequivalent(local)martingalemeasures(E(L)MMs), respectivelyequivalentlocal
martingaledeflators(ELMDs).Moreprecisely,wecharacterizethesetsofE(L)MMs
bysetsofsolutionsofgeneralizedmartingaleproblems,andprovidemilddetermin-
istic conditions for the existence of an ELMD. Employing comparison arguments
and explosion conditions, we derive deterministic sufficient and necessary condi-
tions for the existence of an E(L)MM that are convenient to verify. Moreover, we
Date:November 8,2016.
2010MathematicsSubjectClassification. 60G44,60G48,60H15,60H30,60J25,60J60,91B70.
Keywordsandphrases. (NFLVR),(NUPBR),(NGA),financialbubble,(infinite-dimensional)
diffusions,Heath-Jarrow-Morton-MusielaSPDE,equivalentmartingalemeasure.
D. Criens - Technical University of Munich, Department of Mathematics, Germany,
[email protected].
Acknowledgements: Theauthor thanksKathrinGlauandJanNagelforfruitfuldiscussions.
1
2 D.CRIENS
provide deterministic conditions for the existence of a financial bubble in the sense
of Cox and Hobson [11].
On the other hand, we study a Heath-Jarrow-Morton-Musiela (HJMM) term
structureframework.StartingfromtheclassicalviewofHeath,Jarrow,andMorton
[36], for each maturity T the forward rate process f(,T) is assumed to follow Itoˆ
·
dynamics.Inthis case,Musiela’s [4, 69]parametrizationr(t,x) f(t,t+x)solves,
≡
in the mild sense, the SPDE
dr
(1.1) dr(t,x)= (t,x)+µ(t,x) dt+σ(t,x)dW ,
t
dx
(cid:18) (cid:19)
where W is a cylindrical Brownian motion. For the homogeneous case, we show
thatLipschitz conditions,whicharetypicallyimposedinSPDE settings,imply the
classical notion of no arbitrage (NA).
Theabsenceofarbitrageforaclassof(finite-dimensional)diffusionmodels were
studied by Lyasoff [64], Delbaen and Shirakawa [20], and Mijatovi´c and Urusov
[67].LyasoffworksinamarketdrivenbyanItoˆprocess.Heshowsthat(NFLVR)is
determined by the equivalence of two probability measures, one of them being the
Wienermeasure.Hisresultsdiffersinanimportantaspectfromours.Whilewechar-
acterize explicitly the setof ELMMs as a setof solutions to generalizedmartingale
problems,andtherebyobtainexplicitdeterministiccriteriafor(NFLVR),Lyasoff’s
characterization does not provide deterministic conditions which are suitable for
applications. The spirit of the work of Delbaen and Shirakawa, and Mijatovi´c and
Urusov is close to ours. However,they work in a one-dimensional setting, while we
areparticularlyinterestedinmultidimensionalcases,whichdiffers inmanyaspects
fromtheone-dimensionalone.Toillustratethis,weconstructad-dimensionalmar-
ket which allows for arbitrage opportunities if and only if d 3, c.f. Section 3.1.3.
≥
This is particularly surprising when noting that the sources of risk in this market
have the same dimension.
To derive our characterizations of arbitrage, we need a powerful technical tool
whichisrobustintermsofdimensions.ThistoolisageneralizedGirsanovtheorem,
relating solutions of two generalized martingale problems to a candidate density
process. Thanks to the theorem we can study the two important questions, when
twogeneralizedmartingaleproblemsareequivalent,andwhenthecandidatedensity
process is a martingale. These are exactly those two questions one has to answer
when studying the existence and absence of arbitrage. Our generalized Girsanov
theorem has three interesting byproducts. First, it allows us to give a collection
of sufficient and necessary deterministic conditions for stochastic exponentials to
be strict local, respectively true, martingales. Second, it implies that SPDEs with
differentdriftcoefficientsandthesamediffusioncoefficientanddeterministicinitial
condition are equivalent in law if they have unique solutions for all deterministic
initialconditions.ThisresultseemstobenewforSPDEs,andextendsthediscussion
ofDaPratoandZabczyk[13].Third,itallowsustorelatetheF¨ollmermeasure[30]
tosolutionsofgeneralizedmartingaleproblems.WeprovethegeneralizedGirsanov
theorem by combining a technique based on a local change of measure, introduced
by Sin [84] to the financial literature, and a local uniqueness observations which is
related to the concept of local uniqueness introduced by Jacod [43].
We shortly summarize the structure of the article. In Section 2 we recall the
mathematicalbackgroundofthe financialmodels studied inthis article.In Section
3.1, we state our abstract and our deterministic conditions in the semimartingale-
diffusion framework. In particular, we discuss the one dimensional case in Section
3.1.4.InSection3.2westatedeterministicLipschitzconditionsimplying(NA)inthe
HJMM framework. The proofs are given in Section 4, where Corollary 4.4 relates
the F¨ollmer measures to solutions of generalized martingale problems, Corollary
3
4.5 provides deterministic necessary and sufficient conditions for the martingale
propertyofanon-negativestochasticexponential,andCorollary4.6givessufficient
conditionsfortheequivalenceinlawoftwoSPDEswithdifferentdrift.Wecollected
our notations,as wellas importantobservationsconcerning generalizedmartingale
problems and S(P)DEs, in Appendix A. To be self-contained, at least up to some
level, we have added proofs for of all results in the appendix.
2. Concepts of No Arbitrage in a Semimartingale-Diffusion
Framework
In this section we introduce the two financial markets studied in this article.
For notations, as well as definitions of generalized martingale problems and weak
solution of SPDEs, we refer to Appendix A.
2.1. A Semimartingale-Diffusion Framework.
2.1.1. The Model. We define the real-world measure P of our financial market to
be a solution to the generalized martingale problem (0,σ,µ,x , ) for some x
0 0
H 0 . Denote by (en)n∈K an orthonormal basis of H, where K∞is some countabl∈e
in\d{ex}set whose cardinalityequalsthe Hilbert space dimensionof H.We define the
discounted price process S (Sn)n∈K by
≡
(2.1) Sn ( X⋆,e ), n K,
n
≡E h i ∈
where denotes the stochastic exponential.
E
2.1.2. NotionsofArbitrage. Inasemimartingale-diffusionmarket,theclassicalcon-
cept of no arbitrage is (NFLVR) which was introduced by Delbaen and Schacher-
meyer [14, 18]. For a detailed economical interpretation and an overview on the
historical development we refer to their monograph [19]. A slightly stronger con-
dition is (NGA) as introduced by Cherny [8] and Yan [93]. We refer to Cherny’s
article for a comparison of (NFLVR) and (NGA). Let us, however, point out, that
the difference between (NFLVR) and (NGA) can be economically interpreted as
the absence or existence of a financial bubble as defined by Cox and Hobson [11].
A financial bubble can be seen as an opponent of the classical equilibrium theory.
Moreprecisely,if(NFLVR)holds,while(NGA) fails,thenthe priceoftheunderly-
ingstockis toohighcomparedto the priceoneshouldpay.Manysubtle thingscan
happen due to the occurrence of a financial bubble. For example, Cox and Hobson
notedthatinthepresenceofafinancialbubble manyfolkloreresultsmayfail.This
includes the put-call parity, the fact that the price of an American call exceeds
the price of an European call, and the fact that the call prices are increasing in
its maturity. For a comprehensive overview on the mathematics of financial bub-
bles we refer to the article of Protter [75] and the articles of Jarrow, Protter, and
Shimbo [45, 46]. From the perspective of portfolio optimization, one should not a
priori rule out the possibility of arbitrage. This motivated Karatzas and Kardaras
[53]tostudythe notionof(NUPBR),whichthey provedtobethe minimalapriori
notion to solve problems of portfolio optimization in a meaningful manner. It is
equivalent to (BR) as defined by Kabanov [48] and in the one-dimensional setting
equivalent to the No Arbitrage of the First Kind as studied by Kardaras [56]. The
importanceof(NUPBR)wasalsonoticedbyDelbaenandSchachermayer[16],who
related(NFLVR)to(NUPBR)andthe classicalnotion(NA). Moreover,(NUPBR)
allowsto study optimal arbitrage,c.f. [28, 81],to establisha modified put-call par-
ity,c.f.[81],andto evaluatecall-typeAmericanoptionsin the presenceoffinancial
bubbles, c.f. [1].
Inthisarticlewedefine(NFLVR),(NGA),and(NUPBR)bytheircorresponding
fundamental theorems of asset pricing. The story of the fundamental theorems
4 D.CRIENS
goes back to the seminal work of Harrison, Pliska, and Kreps [33, 34, 59], who
noted that the absence of arbitrage is connected to the existence of an equivalent
martingale measure. For the notion (NFLVR) and generalsemimartingale markets
thisimportantresultwasprovenintheseminalworkofDelbaenandSchachermayer
[14, 18]. For (NGA) the fundamental theorem was given by Cherny [8]. In the case
of(NUPBR),fundamentaltheoremsinvariousdegreesofgeneralitywerediscussed
in the work of Imkeller and Perkowski [42], Karatzas and Kardaras [53], Kardaras
[56], and Schweizer and Takaoka [87].
Let T (0, ] be a time horizon. If T , we identify [0,T] with [0, ). In
∈ ∞ ≡ ∞ ∞
particular,wewriteXT (X ) ,anddenotethefiltrationF ( o) .
≡ t∧T t∈[0,∞) T ≡ Ft t∈[0,T]
Definition 2.1. (i) We call a probability measure Q on (Ωo, o) an equiva-
F
lent local martingale measure (ELMM) on [0,T], if Q P on o and ST
∼ FT
is a local (F ,Q)-martingale. We denote by (T) the set of E(L)MMs.
T (l)
M
(ii) Wecallaprobability measureQon(Ωo, o)an equivalentmartingalemea-
F
sure (EMM) on [0,T], if Q P on o and ST is an uniformly integrable
∼ FT
(F ,Q)-martingale. We denote the set of EMMs by (T).
T
M
(iii) Wecall a probability measureQ on (Ωo, o) a localequivalent(local)mar-
F
tingale measure (LE(L)MM), if Q (t) for all t > 0. We denote the
(l)
∈ M
space of LE(L)MMs by loc.
M(l)
(iv) We call a non-negative local (FP,P)-martingale ZT with ZT = 1 and
T 0
P(ZT >0)=1 an equivalent local martingale deflator (ELMD), if ZTST
T
is alocal(FP,P)-local martingale. Wedenoteby (T)thesetof ELMDs.
T Md
Obviously, if T < we may drop the uniform integrability in the definition of
∞
an EMM. By definition, ( ) loc. Moreover, it can be deduced form [44,
M(l) ∞ ⊂ M(l)
Proposition III.3.8], that for each Q (T) the F -density process of Q w.r.t.
l T
∈ M
P is an ELMD. Note, however,that the ELMD may only be a strict local (F ,P)-
T
martingale,i.e.alocal(F ,P)-martingalewhichisnotrue(F ,P)-martingale,while
T T
the density process is always a true (F ,P)-martingale. We give conditions for
T
such a situation in Section 3.1.1 below. Examples and properties of strict local
martingales can be found in the works [15, 17, 23, 68, 72]. Let us stress that the
definition of a strict local martingale given by Delbaen and Schachermayer [15] is
slightlydifferentfromours,whichistheclassicaldefinitionofElworthy,Li,andYor
[23].Beingmoreprecise,DelbaenandSchachermayerdefineastrictlocalmartingale
to be a local martingale which is no uniformly integrable martingale. In contrast
to the finite time horizon, on the infinite time horizon this definitions does not
coincide with ours. For example, consider the process (B), where B is a one-
E
dimensional Brownian motion. It is easy to see that this process is a martingale
whichisnotuniformlyintegrableasitconvergesa.s.tozeroast .However,we
→∞
note that each strict local martingale in our sense is also a strict local martingale
in the sense of Delbaen and Schachermayer. Takaoka and Schweizer [87] study
the concept of equivalent sigma martingale deflators (ESMDs). In our setting the
sets of ESMDs and ELMDs coincide, since non-negative sigma martingales are
local martingales. The following definitions are in the spirit of the corresponding
fundamental theorems which we note in each definition.
Definition 2.2. (i) [14, Corollary 1.2] We say that (NFLVR) holds on [0,T]
if there exists an ELMM on [0,T], i.e. (T)= .
l
M 6 ∅
(ii) We say that (LNFLVR) holds if there exists a LELMM, i.e. loc = .
Ml 6 ∅
(iii) [8, Theorem 3.6, Lemma 4.1] We say that (NGA) holds on [0,T] if there
exists an EMM on [0,T], i.e. (T)= .
M 6 ∅
(iv) We say that (LNGA) holds if there exists a LEMM, i.e. loc = .
M 6 ∅
5
(v) [87, Theorem 2.6] We say that (NUPBR) holds on [0,T] if there exists an
ELMD on [0,T], i.e. (T)= .
d
M 6 ∅
(vi) We say that the market includes a financialbubble on [0,T], if there exists
a probability measure Q on (Ωo, o) such that Q P on o and ST is a
F ∼ FT
strict local (F ,Q)-martingale.
T
These definitions are in line with the properties we mentioned before, i.e.
(NGA) = (NFLVR),
⇒
and a financial bubble exists if and only if (NFLVR) holds while (NGA) fails.
Remark 2.3. (i) If loc = , i.e. (LNGA) holds, then (NFLVR), (NGA),
M 6 ∅
and (NUPBR) hold on all finite time intervals.
(ii) If loc = , i.e. (LNFLVR) holds, then (NFLVR) and (NUPBR) hold on
Ml 6 ∅
all finite time intervals.
(iii) If loc = , i.e. (LNFLVR) fails, then (NFLVR) and (NGA) fail on the
Ml ∅
infinite time horizon.
2.2. An infinite-dimensional HJMM Framework. In this section we recall
the(homogeneous)infinite-dimensionalHJMMframeworkasintroducedbyMusiela
[4,69]andFilipovi´c[29],whichmodelstheforwardrateasa(mild)solutionprocess
to the SPDE
dr
(2.2) dr(t,x)= (t,x)+α(r(t,x)) dt+ζ(r(t,x))dW ,
t
dx
(cid:18) (cid:19)
where W is a cylindrical Brownian motion. The equation (2.2) is typically called
the HJMM equation.
Remark 2.4. Let us shortly relate the (non-homogeneous)HJMM equation (1.1)
to the classical formulation of Heath, Jarrow, and Morton. We assume that the
dynamic of the forward rare f(,T) for maturity T are given by the Itoˆ dynamics
·
df(t,T)=µ′(t,T)dt+σ′(t,T)dW .
t
Then, we may relate
µ(t,x)=µ′(t,t+x),
σ(t,x)=σ′(t,t+x).
In the case of the Hull and White model [37], for instance, we have σ′(t,T)
≡
σexp( a(T t)), σ,a>0, which simply yields σ(t,x)=σexp( ax).
− − −
From a mathematical point of view, an SPDE approach is typically more chal-
lenging than classical SDE approaches. To name only one reason, mild solution
processesto SPDEsareingeneralno semimartingalesand,therefore,muchofclas-
sical theory is not accessible. However, under typical Lipschitz conditions, mild
solutions are (analytically) weak solutions and we can pose ourselves in a setting
based on generalized martingale problems.
2.2.1. The Model. We startwitha descriptionofaneconomicallymeaningfulstate
space H.
(i) TheelementsofHarecontinuous,real-valuedfunctionswithdomain[0, ).
We assume that the evaluation functional δ g g(x), g H,x [0, )∞, is
x
well-definedasanelement inthe dualofH.Mo≡reover,we∈suppo∈se th∞atthe
right-shift semigroup is strongly continuous on H.
S
6 D.CRIENS
(ii) We are given the measurable map F : D H, where D L (U,H) is
HJM 2
→ ⊂
non-empty, defined by
x
F (y)(x) δ y,I y , y L (U,H),I g g(s)ds,x [0, ).
HJM x x 2 x
≡h i ∈ ≡ ∈ ∞
Z0
(iii) We are given two Borel maps λ: H U and ζ: H D, and a fixed initial
value f H. → →
0
∈
Example. Filipovi´c [29] proposed the following weighted Sobolev space as con-
crete choice of H. For a positive increasing function w: [0, ) [0, ) such
that ∞w(x)−1dx< , let H be the space of all absolutely c∞ontin→uous f∞unctions
f: [0,0 ) R such th∞at
R∞ →
∞
[Df(y)]2w(y)dy < ,
∞
Z0
where Df denotes the weak derivativeof f.Now,endowedwith the scalarproduct
∞
f,g f(0)g(0)+ Df(y)Dg(y)w(y)dy,
h i≡
Z0
H is a real separable Hilbert space, δ is well-defined as an element of the dual
of H, and the right-shift semigroup is strongly continuous on H, c.f. [6, 29]. In
particular, if the weight w satisfiesSthe growth condition ∞y2w−1(y)dy < ,
then F is Lipschitz continuous on bounded subsets from0L (U,Ho) into Ho∞
HJM 2
f H such that f( )=0 , c.f. again [6, 29]. For a relatedRexample c.f. e.g. [89≡].
{ ∈ ∞ }
The operator A d/dx is the generator of the right-shift semigroup . Let
≡ S
((Ω, ,F,P);W;r)beatripletconsistingofafilteredprobabilityspace(Ω, ,F,P)
F F
which satisfies the usual conditions, and supports a cylindrical Brownian motion
W and an H-valued continuous process r so that
t t
P r = r + α(r )ds+ ζ(r )dW =1, t [0, ),
t t 0 t−s s t−s s s
S S S ∀ ∈ ∞
(cid:18) Z0 Z0 (cid:19)
whereα F ζ ζλ.Inotherwords,((Ω, ,F,P);W;r)isaso-calledmartingale
HJM
≡ − F
solution to the SPDE (A,ζ,α,r , ), c.f. e.g. [13, 31, 63]. We additionally assume
0
thatζ,λ,andαareboundedonbo∞undedsubsetsofH,andthat((Ω, ,F,P);W;r)
F
is also a weak solution to the S(P)DE (A,ζ,α,r , ). Mild and weak solutions of
0
∞
SPDEsarecloselyrelated,c.f.[63,AppendixG].UndertypicalLipschitzandlinear
growthconditions,any mild solution is also a weak solution, c.f. e.g. [88, Corollary
53].
We define the money market account B by
·
B exp r (0)ds ,
s
≡
(cid:18)Z0 (cid:19)
and the bond price processe P by
x
P(t,x) exp r (y)dy , t,x [0, ).
t
≡ − ∈ ∞
(cid:18) Z0 (cid:19)
2.2.2. Notion of Arbitrage. The notion of arbitrageused in this HJMM framework
is the classical no arbitrage (NA) notion as introduced by Harrison and Pliska
[34]. Moreover, the easy implication of the fundamental theorem also holds in this
setting, c.f. e.g. [6, Theorem 2.1].
Definition 2.5. We say that (NA) holds on [0,T], if there exists a probability
measure Q on (Ω, ) which is equivalent to P such that for all T⋆ [0,T] the
F ∈
discounted price process (B−1P(t,T⋆ t)) is a local (F,Q)-martingale.
t − t∈[0,T⋆]
7
The question when (NA) holds for this infinite-dimensional HJMM framework
wassystematicallystudiedbyFilipovi´c[29].HeimposedNovikov-typeintegrability
conditionsthatareingeneralhardtoverify.InSection3.2wederiveLipschitz-type
conditions that replace these exponential moment conditions.
3. Main Results
In this section we present our main contributions. In Section 3.1 we study the
existence and absence of arbitrage in the sense of (NUPBR), ((L)NFLVR) and
((L)NGA),aswellastheexistenceandabsenceoffinancialbubbles,forthe(infinite-
dimensional) semimartingale-diffusion setting introduced in Section 2.1. In Sec-
tion 3.2 we give sufficient conditions for (NA) in the HJMM frameworkintroduced
in Section 2.2.
3.1. Conditions for (No) Arbitrage in a Semimartingale Framework. We
consider the framework introduced in Section 2.1, and denote a σσ∗.
≡
3.1.1. Conditions for (NUPBR) and ((L)NFLVR). The intuitive candidate for an
ELMD in our continuous setting is Z given by
· ·
(3.1) Z (a−1µ)(X⋆),dX⋆ , X⋆ X⋆ µ(X⋆)ds.
≡E − h s si ≡ − s
(cid:18) Z0 (cid:19) Z0
Under fairlymildconditions,giveninfoellowingeproposition,Z is indeedanELMD.
For a proof we refer to Section 4.2 below.
Proposition 3.1. Assume that a−1 exists and that a−1µ is bounded on bounded
subsets of H, then (NUPBR) holds on any finite time interval.
ManyofthefollowingconditionsarebasedontheassumptionsofProposition3.1.
Letusshortlygivetheintuitionbehindourcharacterizationsof(NFLVR)which
we present below. Recalling the classical Girsanov theorem, by a locally equiva-
lent change of probability measure, only the drift of X⋆ can be influenced. Hence,
since continuous local martingales with bounded variation are constant up to in-
distinguishability, a LELMM has to be a solution to the generalized martingale
problem (0,σ,0,x , ). Conversely, a solution to the generalized martingale prob-
0
∞
lem (0,σ,0,x , ) is an LELMM if it is locally equivalent to P, which holds true
0
∞
under mild deterministic conditions thanks to Proposition 4.1 below. A similar ar-
gumentation allows for a description of the set of ELMMs on finite time intervals.
Besides these characterizations of the sets of LELMM and ELMMs in terms
of generalized martingale problems, we ask for classical characterizations in terms
of the martingale property of the ELMD Z given by (3.1). For finite T, if Z is
an (Fo,P)-martingale, then we can define a probability measure Q by the Radon-
NikodymdensityZ andtheclassicalGirsanovtheoremyieldsthatQisanELMM
T
on [0,T]. In fact, the converse direction also holds true, which we deduce from
Proposition 4.1 below.
Ontheinfinite time horizonthe situationistypicallymoredelicateandonetyp-
ically has to assume that the candidate density process is a uniformly integrable
martingale.Inourcase,webenefitfromthefactthatthefilteredspace(Ωo, o,Fo)
F +
allows extensions of consistent families of probability measures,which implies that
we may define a (locally) equivalent measures in the case where Z is only a mar-
tingale. Employing this fact and again the classical Girsanov theorem, one obtains
that Z being an (Fo,P)-martingale implies the existence of a LELMM. As on the
finite time horizon, also the converse holds, i.e. if a LELMM exists, then Z is an
(Fo,P)-martingale.
To show all these claims we assume the following mild structural assumptions
on the coefficients µ and σ.
8 D.CRIENS
Condition 3.1. Assume one of the following conditions.
(i) Suppose that H Rd, that a is continuous, that
≡
(3.2) θ,a(x)θ >0, x Rd,θ Rd 0 ,
h i ∀ ∈ ∈ \{ }
and that a−1µ is bounded on bounded subsets of Rd.
(ii) Suppose that a−1 exists, a−1µ is bounded on bounded subsets of H, and that
σ is Lipschitz continuous on bounded subsets of H.
Notethatthefirstpartof (3.2)impliesthata−1exists.Part(i)ofCondition3.1is
relatedtotheEngelbert-Schmidtconditions[24,25]intheone-dimensionaldiffusion
setting, i.e. for d = 1 the first part translates to a(x) > 0 for all x R, and the
continuity of a implies a−1 L1 (R). The first main result of this s∈ection is the
∈ loc
following.
Theorem 3.2. Assume that Condition 3.1 holds. Then
loc = (0,σ,0,x , ), (T)= (0,σ,0,x ,T),
Ml IG 0 ∞ Ml IG 0
and loc 1. Moreover,
|Ml |≤
loc = Z is an (Fo,P)-martingale,
(3.3) Ml 6 ∅ ⇐⇒
(T)= ZT is an (F ,P)-martingale,
l T
M 6 ∅ ⇐⇒
where Z is given as in (3.1). In particular, if the generalized martingale problem
(0,σ,0,x , ) has a solution, then loc = 1, (LNFLVR) holds, and (NFLVR)
0 ∞ |Ml |
and (NUPBR) hold on all finite time intervals. Moreover, (NFLVR) holds on [0,T]
if and only if the generalized martingale problem (0,σ,0,x ,T) has a solution.
0
For a proof of this theorem we refer to Section 4.3 below. In other worlds, (3.3)
states that (LNFLVR) holds if and only if the ELMD Z is a true martingale.
Remark 3.3. Note that the inclusions
(0,σ,0,x , ) loc, (0,σ,0,x ,T) (T),
IG 0 ∞ ⊂Ml IG 0 ⊂Ml
are also true if (ek)k∈K is no basis.
Next,weaimfordeterministic conditions.Inaone-dimensionaldiffusionsetting
such conditions are given by Mijatovi´c and Urusov [67]. In Section 3.1.4 below we
compare their results to ours.
Let us draw the readers attention on a difficulty which arises on the way to
deterministic conditionsforthe existence ofarbitrageonthe finite time horizon.In
viewofTheorem3.2,(NFLVR) failson[0,T]ifthegeneralizedmartingaleproblem
(0,σ,0,x ,T) has no solution. Let us suppose that (τ < ) > 0, where
0 ∆
Q ∞ Q ∈
(0,σ,0,x ,τ ), then it is a priori not clear that (τ T) > 0 for finite T.
0 ∆ ∆
IG − Q ≤
Hence, we can only conclude that (NFLVR) fails on the infinite time horizon, but
notnecessarilyonanyfinitetimeinterval.However,especiallyfromanapplications
point of view, we are naturally interested in any finite time interval. In fact, under
mild conditionsonthe diffusion coefficient,it wasrecentlyprovenby Karatzasand
Ruf [54] in the case where H R, that (τ < ) > 0 implies (τ T) > 0
∆ ∆
≡ Q ∞ Q ≤
for all T > 0. This result was extended by Bruggeman and Ruf [5] to a more
general one-dimensional diffusion setting. Employing a comparison argument and
thereby reducingour multi- andinfinite-dimensionalcase to one dimension,we use
the result of Karatzas and Ruf to prove Proposition A.21 in Appendix A, which
providesdeterministic conditions for arbitrarilyfastexplosion.These deterministic
conditions allow us to deduce the following corollary from Theorem 3.2.
Corollary 3.4. Suppose that Condition 3.1 holds.
9
(i) Assume that σ is of linear growth, or that Condition A.2 in Appendix A
holds with µ 0 . Then loc =1, (LNFLVR) holds, and (NFLVR) and
∈{ } |Ml |
(NUPBR) hold on all finite time intervals.
(ii) Assume that Condition A.1 holds with µ 0 .
∈{ }
(ii.a) If additionally (A.15) in Appendix A holds, then for T > 0 large
enough, (T) = , and (NUPBR) holds, while (NFLVR) fails on
l
M ∅
[0,T].
(ii.b) If additionally w1/2 is locally of h-class, wv is locally Lipschitz con-
tinuous, (A.15) and (A.16) in Appendix A hold, then for all T > 0,
(T)= , and (NUPBR) holds, while (NFLVR) fails on [0,T].
l
M ∅
3.1.2. Conditions for (NGA). To motivate this section, let us have a short look
on the special situation where µ 0. In this case, P is already an ELMM and
≡
(NFLVR) and (NUPBR) hold. However,S might not be a true (F ,P)-martingale
T
and(NGA)canfail.Inotherwords,thefinancialmarketmaystillincludeafinancial
bubble which influences many folklore results. To be aware of the consequences of
the existence of a financial bubble, it is important to know whether (NGA) holds
or fails. The main result of this section is given by the following.
Theorem 3.5. Assume that Condition 3.1 holds. Then
(0,σ,0,x , )= ,
(3.4) loc = IG 0 ∞ 6 ∅
M 6 ∅ ⇐⇒ ( (0,σ,σσ∗en,x0, )= , n K.
IG ∞ 6 ∅ ∀ ∈
Moreover, loc 1, and
|M |≤
(0,σ,0,x ,T)= ,
0
(3.5) (T)= IG 6 ∅
M 6 ∅ ⇐⇒ ( (0,σ,σσ∗en,x0,T)= , n K.
IG 6 ∅ ∀ ∈
In particular, if for all n K the generalized martingale problems (0,σ,0,x , )
0
∈ ∞
and (0,σ,σσ∗e ,x, ) (resp. (0,σ,σσ∗e ,x ,T) and (0,σ,σσ∗e ,x,T)) are well-
n n 0 n
∞
posed, then loc = 1 (resp. (T) = 1), (LNGA) holds, (NGA) holds and no
|M | |M |
financial bubble exists on all finite time intervals (resp. on [0,T]). Moreover, if
the generalized martingale problem (0,σ,0,x ,T) has a solution, while for some
0
n K the generalized martingale problem (0,σ,σσ∗e ,x ,T) has no solution, then
n 0
∈
(NFLVR) holds, while (NGA) fails on [0,T]. In this case a financial bubble exists
on [0,T].
For a proof we refer to Section 4.4 below. We now give explicit deterministic
conditions, which are consequences of Theorem 3.5 and the explosion conditions
given by Proposition A.21 in Appendix A.
Corollary 3.6. Suppose that Condition 3.1 holds, and that σ is of linear growth,
or that Condition A.2 in Appendix A holds with µ 0 .
∈{ }
(i) In addition, assume that either for all n K, σσ∗e is of linear growth,
n
or that Condition A.2 in Appendix A hold∈s with µ σσ∗e ,n K , then
n
∈{ ∈ }
loc =1, (LNGA) holds, (NGA) holds and no financial bubble exists on
|M |
all finite time intervals.
(ii) In addition, assume that for some n K, Condition A.1 and (A.15) in
∈
Appendix A hold for µ σσ∗e , then for T >0 large enough (T)= ,
n
∈{ } M ∅
(NFLVR) holds, while (NGA) fails, and a financial bubble exists on [0,T].
(iii) In addition, assume that for some n K, Condition A.1 holds, w1/2 is
∈
locally of h-class, wv is locally Lipschitz continuous, (A.15) and (A.16) in
AppendixAholdforµ σσ∗e ,thenforallT >0, (T)= ,(NFLVR)
n
∈{ } M ∅
holds, while (NGA) fails, and a financial bubble exists on [0,T].
In the following section we restrict ourselves to a finite-dimensional setting.
10 D.CRIENS
3.1.3. The Influences of the Market Dimension. The number of assets in a market
playsa crucialrolefor the decisionwhether (NFLVR) holds or fails.Inthis section
wegiveanexampleforafinancialmarketinwhichthe dimensionsofthe sourcesof
risk coincide, but depending onthe number of assets,(NFLVR) holds, respectively
fails.
Let us suppose that H= Rd, the coefficient a is given by a a⋆1 , where
ij {i=j}
≡
a⋆: Rd (0, ) is continuous and such that
→ ∞
(3.6) inf a⋆(x)>0, for all R>0.
|x|≤R
In the case d 3, by using v(z) = dz−1 and w(z) = 2zα(√2z), we translate
≥ 2
Condition A.1 and (A.15) in Appendix A to the following.
Condition3.2. Assumethatthereexistsacontinuousfunctionα: (0, ) (0, )
∞ → ∞
such that α(ρ) a⋆(x) for ρ>0 and x =ρ, and
≤ | |
∞ 1
(3.7) dρ< .
α(√ρ) ∞
Z1
Ifd=1,2,thenthegeneralizedmartingaleproblem(0,σ,0,x , )iswell-posed,
0
∞
while if d 3 and Condition 3.2 holds, then it has no solution, c.f. [86, Exercise
≥
10.3.3]. These observations yield the following corollary of Theorem 3.2.
Corollary 3.7. Inthesettingdescribed above, if d 2,then (LNFLVR)holds, and
≤
(NFLVR) holds on all finite time intervals. If d 3 and Condition 3.2 holds, then
≥
(NFLVR) and (NGA) fail on the infinite time horizon and for T >0 large enough
(NFLVR) and (NGA) fail on [0,T].
As we will see in the next section, in contrastto (NFLVR), the current assump-
tionsdonotimplythat(NGA)holdsford=1,2.Wecanalsogiveaconditionsuch
that (NFLVR) and (NGA) fail on arbitrarily small finite time intervals.
Corollary 3.8. In the setting described above, if d 3 and Condition 3.2 holds
≥
with α being locally Lipschitz continuous, then (NFLVR) and (NGA) fail on [0,T]
for all T >0.
Proof: RecallthatConditionA.1 and(A.15)inAppendix A,andCondition3.2are
related by v(z) = dz−1/2 and w(z) = 2zα(√2z). Now, since α is locally Lipschitz
continuous, so are w1/2 and vw. Moreover,since d 3, we have
≥
1/2 z 1/2 1
(3.8) exp v(u)du dz =const. dz =+ .
Z0 −Z1/2 ! Z0 zd/2 ∞
Hence, in view of [55, Problem 5.5.27], the claim follows from Corollary 3.4. (cid:3)
Note that (3.8) indicates the influence of the dimension in (A.15). We now turn
to the one-dimensional case.
3.1.4. The One-Dimensional Case. In the case H R we have already seen that
≡
(NFLVR) holds typically. Now, we give also very sharp conditions for (NGA). Let
us furthermore stress that all conditions are considerablyweaker than the classical
exponential moment conditions of Novikov-type which are typically imposed to
assure (NFLVR) in semimartingale settings, c.f. e.g. [33, 52, 76].
Corollary 3.9. Suppose that a is continuous, and that (3.6) holds with a⋆ replaced
by a. If it holds that
∞ 1
(3.9) dy =+ ,
a(y) ∞
Z1